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APPLICATIONS OF LIE THEORY

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APPLICATIONS OF LIE THEORY

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email matwml@nus.edu.sg

Tel (65) 6516-2749

http://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2006/USC3002

THE UTILITY OF LIE THEORY

Lie theory models nonlinear constraints and symmetry

that explains phenomena and arises in engineering

design and simulation of physical processes.

Shlomo Sternberg - Group Theory and Physics

1928 Weyl’s Gruppentheorie und Quantenmechanik

1920-30’s Chemistry and Spectroscopy

1930-40’s Nuclear and Particle Physics

1960-70’s High Energy Physics

THE GRUPPENPEST SYNDROME

Sternberg notes: the remarks on pages 60-2 in Slater, J.C.Solid-State and

Molecular Theory: A Scientific Biography. New York: Wiley, 1975.

“It was at this point that Wigner, Hund, Heitler, and Weyl

entered the picture with their “Gruppenpest”: the pest of group

theory…The authors of “Gruppenpest” wrote papers that were

incomprehensible to those like me who had not studied group

theory, in which they applied these theoretical results to the

study of the many electron problem. The practical

consequences appeared to be negligible, but everyone

felt that to be in the mainstream one had to learn about it.”

and remarks “It is, however, amazing to consider that this

autobiography was published in 1975, after the major

triumphs of group theory in elementary particle physics.”

RECOGNITION

of the utility of Lie theory in engineering design and physical

simulation may be slower than it was in physics, due to

1. the extensive mathematical background of physicists, and

Slater, John Clarke (1900-1976) American physicist who did work on the application of quantum mechanics to the chemical bond and the structure of substances. His name is attached to the Slater determinant used to construct antisymmetric wavefunctions.

2. the fundamental nature of physics. But the success of Lie

theory in physics may facilitate recognition in other fields.

Calderbank and Moran have demonstrated the importance

of discrete Heisenberg-Weyl groups in coding.

Lie symmetry explains soliton/instanton phenomena that have

growing importance in material science and nanotechnology.

THREE APPLICATIONS

were chosen based on 1. my interests and 2. instructive value.

Functions with values in Lie groups

Filter Design

Dynamic Simulation

Functions (measures) on Lie groups

Localization Explanation

(for waves in random media)

sequence

whose Fourier transform

CONJUGATE QUADRATURE FILTER

or frequency response

defined by

satisfies the nonlinear constraint

where

and

Definition F has regularity N if

Regular CQF’s used for filterbanks & wavelets

is represented by a polyphase vector

CONJUGATE QUADRATURE FILTER

and there exists a

where

polyphase matrix

whose 1st column equals

Theorem 1.

is a trigonometric polynomial (finite

impulse response filter) iff every component of

is. Then

can be chosen to have trig. poly. entries.

Lemma 1.

(in Pressley and Segal’s Loop Groups)

CONJUGATE QUADRATURE FILTER

Every continuous

can be uniformly

approximated by a trig. poly.

Proof. Trotter formula and semisimplicity of SU(m)

Theorem 2.

Every continuous CQF

can be uniformly approximated by a trig. poly. CQF

with the same regularity as

Proof. [1], uses Lemma 1, Jets, and the Brower degree

Euler 1765

GEODESIC FLOW

Flow is a trajectory

with values

in the group of volume-preserving diffeomorphisms.

Velocity

is the angular velocity in space

Moreau 1959 g is a geodesic with respect to the

right-invariant kinetic-energy Riemannian metric

For d = 2, div u = 0 a Stream function

with

where

SDiff(D) is the Lie algebra with Poisson bracket

Vorticity

Flow preserves Casimirs

For periodic flow basis

Euler equations

Quantum deformations

gives algebra derivations

algebra (non comm. torus)

generated by A,B where

For

,N odd gives homomorphism onto

with real form

yielding the

Sine-Euler Approximation by GF on

N-Casimirs are exactly conserved ! See [2]

WAVE PROPAGATION

in a chain of harmonic oscillators

is described by the equation

WAVE PROPAGATION

admits an expansion in normal modes

where

or

and

are samples of a measure on

the Lie group

Theorem Localization of

as

increases.

Proof Furstenberg [3] the Lyapunov exponent

REFERENCES

[1] W. Lawton, Hermite interpolation in loop groups

and conjugate quadrature filter approximation, Acta

Applicandae Mathematicae,84(3),315--349, Dec. 2004

[2] V. Arnold and B. Khesin, Topological Methods

in Hydrodynamics, Springer, New York, 1998.

[3] H. Furstenberg, Noncommuting random products,

Transactions of the AMS, 108, 377-429, 1963.